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A050322
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Number of factorizations indexed by prime signatures: A001055(A025487).
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9
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1, 1, 2, 2, 3, 4, 5, 7, 5, 7, 9, 12, 11, 11, 16, 19, 21, 15, 29, 26, 30, 15, 31, 38, 22, 47, 52, 45, 36, 57, 64, 30, 77, 98, 67, 74, 97, 66, 105, 42, 109, 118, 92, 109, 171, 97, 141, 162, 137, 165, 56, 212, 181, 52, 198, 189, 289, 139, 250, 257, 269, 254, 77, 382, 267
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OFFSET
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1,3
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COMMENTS
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LINKS
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EXAMPLE
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The a(1) = 1 through a(11) = 9 factorizations:
{} 2 4 6 8 12 16 24 30 32 36
2*2 2*3 2*4 2*6 2*8 3*8 5*6 4*8 4*9
2*2*2 3*4 4*4 4*6 2*15 2*16 6*6
2*2*3 2*2*4 2*12 3*10 2*2*8 2*18
2*2*2*2 2*2*6 2*3*5 2*4*4 3*12
2*3*4 2*2*2*4 2*2*9
2*2*2*3 2*2*2*2*2 2*3*6
3*3*4
2*2*3*3
(End)
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MAPLE
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MATHEMATICA
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c[1, r_] := c[1, r] = 1; c[n_, r_] := c[n, r] = Module[{d, i}, d = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/d[[i]], d[[i]]], {i, 1, Length[d]}]]; Map[c[#, #] &, Union@ Table[Times @@ MapIndexed[If[n == 1, 1, Prime[First@ #2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, Product[Prime@ i, {i, 6}]}]] (* Michael De Vlieger, Jul 10 2017, after Dean Hickerson at A001055 *)
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Length/@facs/@First/@GatherBy[Range[1000], If[#==1, {}, Sort[Last/@FactorInteger[#]]]&] (* Gus Wiseman, Jan 13 2020 *)
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CROSSREFS
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The version indexed by unsorted prime signature is A331049.
Cf. A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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